TI-84 Plus CE and Numworks: System of Linear Differential Equations (Simple System)

TI-84 Plus CE and Numworks: System of Linear Differential Equations (Simple System) 

Solving  The System

The program presented today will solve the following system of equations:

dx/dt = A * x(t) + B * y(t)

dy/dt = C * x(t) + D * y(t)

with initial conditions x(0) = x0 and y(0) = y0

The formulas used in the program are derived using the Laplace Transform.

dx/dt = A * x(t) + B * y(t)

dy/dt = C * x(t) + D * y(t)

Laplace Transforms to be used:

ℒ{f(t)} = F(s)

ℒ{df/dt} = s * F(s) - f(0)

Apply the Laplace Transform transform:

s * X(s)  - x0 = A * X(s) + B * Y(s)

s * Y(s) - y0 = C * X (s) + B * Y(s)

(s - A) * X(s) + (-B) * Y(s) = x0

(-C) * X(s) + (s - D) * Y(s) = y0

Define three matrices:

Md = [ [ s - A, -B ], [ -C, s - D ] ]

where

det(Md) = s^2 - (A + D) * s + (A * D - B * C)

which can be factored into:

(s - P) * (s - Q)

where:

P = (( A + D ) + √(( A + D)^2 - 4 * (A * D - B * C)) /2

Q = (( A + D ) - √(( A + D)^2 - 4 * (A * D - B * C)) /2

Mx = [ [ x0, -B ], [ y0, s - D ] ] 

where

det(Mx) = x0 * (s - D) + B * y0

My = [ [ s - A, x0 ], [ - C, y0 ] ]

where 

det(My) = (s - A) * y0 + C * x0

Per Cramer's Rule, the solution to X(s) and Y(s) are:

X(s) = det(Mx) / det(Md)

Y(s) = det(My) / det(Md)

What happens next depends of the roots P and Q.  We're assuming that P and Q are real here but the situation applies if P and Q are complex. 

If P ≠ Q:

If P ≠ Q, then after simplifying and finding partial fractions:

X(s) = ( x0 * (s - D) + B * y0 ) / ((s - P) * (s - Q)) = α / (s - P) + β / (s - Q)

where

α = ( B * y0 + (P - D) *x0) / (P - Q)

β = x0 - α

Y(s) = ( (s - A) * y0 + C * x0 ) / ((s - P) * (s - Q)) = γ / (s - P) + δ / (s - Q)

where

γ =(C * x0 + y0 * (P - A)) / (P - Q)

δ = y0 - γ

Applying the inverse Laplace transform:

x(t) = α * e^(P * t) + β * e^(Q * t)

y(t) = γ * e^(P * t) + δ * e^(Q * t)

If P = Q:

If P = Q, then after simplifying and finding partial fractions:

X(s) = ( x0 * (s - D) + B * y0 ) / (s - P)^2 = α / (s - P)^2 + β / (s - Q)

where

α = B * y0 + (P - D) * x0 

β = x0

Y(s) = ( (s - A) * y0 + C * x0 ) / (s - P)^2 = γ / (s - P)^2 + δ / (s - Q)

where

γ =C * x0 + y0 * (P - A)

δ = y0

Applying the inverse Laplace transform:

x(t) = α * t * e^(P * t) + β * e^(Q * t)

y(t) = γ * t * e^(P * t) + δ * e^(Q * t)

TI-84 Plus CE Program:  LINSYS1

Download the program here:  https://drive.google.com/file/d/14zYpT4xhLRGVvBNriOacIOGBpqXLWswR/view?usp=share_link

Program Listing:

Radian

a+bi

ClrHome

Disp "DX/DT=AX+BY DY/DT=CX+DY"

Prompt A,B,C,D

Input "X0? ",M

Input "Y0? ",N

((A+D)+√((A+D)²-4*(A*D-B*C)))/2→P

((A+D)-√((A+D)²-4*(A*D-B*C)))/2→P

If P≠Q

Then

(B*N-D*M+P*M)/(P-Q)→R

M-R→S

(C*M-A*N+P*N)/(P-Q)→U

N-U→V

ClrHome

Output(2,1,"X=Re^(PT)+Se^(QT)")

Output(3,1,"R= "+toString(R))

Output(4,1,"P= "+toString(P))

Output(5,1,"S= "+toString(S))

Output(6,1,"Q= "+toString(Q))

Pause

ClrHome

Output(2,1,"Y=Ue^(PT)+Ve^(QT)")

Output(3,1,"U= "+toString(U))

Output(4,1,"P= "+toString(P))

Output(5,1,"V= "+toString(V))

Output(6,1,"Q= "+toString(Q))

Pause

Else

B*N-D*M+M*P→P

M→S

C*M-A*N+N*P→U

N→V

ClrHome

Output(2,1,"X=RTe^(PT)+Se^(PT)")

Output(3,1,"R= "+toString(R))

Output(4,1,"P= "+toString(P))

Output(5,1,"S= "+toString(S))

Pause

ClrHome

Output(2,1,"Y=UTe^(PT)+Ve^(PT)")

Output(3,1,"U= "+toString(U))

Output(4,1,"P= "+toString(P))

Output(5,1,"V= "+toString(V))

Pause

End

ClrHome

Python:  dfsys1.py

Link from my Numworks page:  https://my.numworks.com/python/ews31415/dfsys1

Modules: 

cmath:  complex math

mathplotlib.pyplot:  screen plotting and text placement

Script listing:  

from cmath import *

from matplotlib.pyplot import *

# 2023-01-28 EWS

# parameter input

print("systems of differential\nequations")

print("dx/dt=a*x+b*y")

print("dy/dt=c*x+d*y")

a=float(input("a? "))

b=float(input("b? "))

c=float(input("c? "))

d=float(input("d? "))

m=float(input("x0? "))

n=float(input("y0? "))

# characteristic roots

p=((a+d)+sqrt((a+d)**2-4*(a*d-b*c)))/2

q=((a+d)-sqrt((a+d)**2-4*(a*d-b*c)))/2

# set up result screen

axis((0,10,0,10))

axis("off")

# determine solutions

# best used for real solutions

if p!=q:

  r=(b*n+m*(p-d))/(p-q)

  s=m-r

  u=(c*m+n*(p-a))/(p-q)

  v=n-u

  text(0,9,"x=re**(pt)+se**(qt)")

  text(0,8.5,"r="+str(r))

  text(0,8,"p="+str(p))

  text(0,7.5,"s="+str(s))

  text(0,7,"q="+str(q))

  text(0,6,"y=ue**(pt)+se**(vt)")

  text(0,5.5,"u="+str(u))

  text(0,5,"p="+str(p))

  text(0,4.5,"v="+str(v))

  text(0,4,"q="+str(q))

else:

  r=(b*n+m*(p-d))

  s=m

  u=(c*m+n*(p-a))

  v=n

  text(0,9,"x=rte**(pt)+se**(pt)")

  text(0,8.5,"r="+str(r))

  text(0,8,"p="+str(p))

  text(0,7.5,"s="+str(s))

  text(0,6,"y=ute**(pt)+se**(vt)")

  text(0,5.5,"u="+str(u))

  text(0,5,"p="+str(p))

  text(0,4.5,"v="+str(v))

show()

Examples

Example 1

dx/dt = 3 * x(t) - 2 * y(t)

dy/dt = 2 * x(t) - 2 * y(t)

x(0) = 6, y(0) = 9

Solutions:

x(t) = 2 * e^(2*t) + 4 * e^(-t)

y(t) = e^(2*t) + 8 * e^(-t)

Example 2

dx/dt = 3 * x(t) - 4 * y(t)

dy/dt = x(t) - y(t)

x(0) = 1, y(0) = 0

Solutions:

x(t) = 2 * t * e^(t) + e^(t)

y(t) = t * e^(t) 

Eddie 

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP 32S II Statistical Formulas

HP 32S II Statistical Formulas

Statistics Formulas

The 32S II can store formulas for evaluations, including formulas involving stat variables.  

Covariance:

C = r × sx × sy

Root Mean Square Based on X Variable Statistics:

R = SQRT( Σx^2 ÷ n )

Z-Score Based on X Variable Statistics:  

Z = (X - x-bar) ÷ sx

Accessing Stat variables:

r:  [ right shift ]  [ LN ] (L.R.) [ y^x ] (r)

sx:  [ right shift ] [ 1/x ] (s, σ), [ √x ] (sx)

sy:  [ right shift ] [ 1/x ] (s, σ), [ LN ] (sy)

Σx^2:  [ right shift ] [ Σ+ ] (SUMS) [ e^x ] (x)

n:  [ right shift ] [ Σ+ ] (SUMS) [ √x ] (n)

x-bar:  [ right shift ] [ y^x ] (x-bar,y-bar) [ √x ] (x-bar)

Example

Data:  (x, y)

(10.4, 20)

(13.5, 18)

(16.8, 15)

(19.1, 13.9)

Enter statistical data:  y [ENTER] x [ Σ+ ]

Some stats:

x-bar = 14.95

y-bar = 16.725

r = -9.9503133E-1

Covariance:  Solve for C

C = -10.555

Result is stored in C

Root Mean Square:  Solve for R

R = 15.3089842903

Result is stored in R

Z-Score Based on X Variable Statistics:   Solve for Z

X?  15,  Z = 1.31382E-2

X?  17,  Z = 5.38666E-1

Result is stored in Z

Formulas vs. Programs

If we are going to evaluate formulas (that is, solve for the variable left of the equals sign), we could also enter and use programs.   The programs for the above three formulas:

Covariance:

C01  LBL C

C02  r

C03  sx

C04  ×

C05  sy

C06  ×

C07  RTN

Root Mean Square:

R01  LBL R

R02  Σx^2

R03  n

R04  ÷

R05  SQRT

R06  RTN

Z Score:  (x is in the X stack)

Z01  LBL Z

Z02  x-bar

Z03  -

Z04  sx

Z05  ÷

Z06  RTN

Byte Consumption Comparison

Covariance

Formula:  12 bytes

Program:  10.5 bytes

Root Mean Square

Formula:  12 bytes

Program:  9 bytes

Z Score

Formula: 15 bytes

Program:  9 bytes

From sample set, programs are shorter than formulas.  Is this true in general?

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP 10bII+ and Sharp EL-W516T: Normal Distribution Function

HP 10bII+ and Sharp EL-W516T:  Normal Distribution Function

Sometimes scientific calculators and financial calculators have normal distribution functions.   

HP 10bII+  Normal Distribution

On the HP 10bII+ financial calculator, the normal distribution functions are available at any time:

[ blue shift ] ( z<>p ):   lower tail area of normal distribution [ P(x) ]

[ blue shift ] ( INV) ( z<>p ):  given area, get the probability

Example:

P(0.2) returns 0.57926

If the area is 0.6, z is 0.253347

Sharp EL-W516T Normal Distribution

The normal distribution functions offered on the Sharp EL-W516T are:

[ MATH ] menu in SD mode:

0) →t:   convert to z

1)  P(:  lower tail area

2)  Q(:  I do believe it is the area between the mean and z

3)  R(:  upper tail area

The normal distribution functions are available only in the statistics mode.  

Example:

P(0.2) returns 0.57296.

Is Statistical Data Required?

No.  In general, especially the normal distribution functions does not require or depend on any stat data.  

However, you use the stat data points to convert from x-values to convert stats to a z value in the EL-W516T using →t.  Here is an example:

Data set:  -3, -1, 2, 5, 6 

Convert the data point 0 and then find the lower tail area.

0→t=   -0.524890659

P(ANS) = 0.29983

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP 12C: An Improved Linear Regression Function (from Curvee RPN-45 SD)

HP 12C:  An Improved Linear Regression Function (from Curvee RPN-45 SD)

Curvee RPN-45 Simulator 

The Curvee RPN-45 from Curvee Software is an excellent app.  The RPN-45 SD app is the Super-45.   The Super-45 is a emulator of the classic HP 45 calculator from 1973 with the following enhancements:

*  The gold shift key becomes a double-shift key (gold/purple)

*  Complex Numbers

*  10 additional registers

*  Days between dates

*  Four solvers: 2 x 2 system of linear equations, 3 x 3 system of linear equations, quadratic equation, cubic equation

*  Expanded statistics including full linear regression analysis

I have purchased one on my iPod Touch (that is an iPhone without the "phone" part) for $1.99 (prices may vary).  There are versions for the iPhone and iPad.  

For more information:  http://cuveesoft.ch/rpn45/en/index.html

The enhanced linear regression (L.R.) function on the Super-45 returns the following calculations to the four-level stack:

T:  covariance

Z:  correlation

Y:  slope

X:  intercept

The relationship between correlation (r) and covariance (cov):

cov(x,y) = r * sx * sy

sx = standard deviation of x-data

sy = standard deviation of y-data

The HP 12C mimics the enhanced L.R. function.   Yes, check out this app!

HP 12C Program:  Enhanced Linear Regression

The results are stored in the following registers:

R7 = intercept

R8 = Slope

R9 = correlation

R.0 = covariance

In addition, I added code to calculate population deviation:  σx in the x-stack, σy in the y-stack.

Step #: Code [ key ]

// linear regression function 

01:  1  [ 1 ]

02:  43, 1  [ x^,r ]

03:  34  [ x<>y ]

04:  44, 9 [ STO 9 ]

05:  43, 48  [ s ]

06:  20  [ × ]

07:  20  [ × ]

08:  44, 48, 0   [ STO  .  0  (decimal point, zero) ]

09:  0  [ 0 ]

10:  43, 2  [ y^, r ]

11:  44, 7  [ STO 7 ]

12:  1   [  1 ]

13:  43, 2  [ y^, r ]

14:  34  [ x<>y ]

15:  33  [ R↓ ]

16:  34  [ x<>y ]

17:  30  [ - ]

18:  44, 8 [ STO 8 ]

19:  45, 48, 0 [  RCL . 0  (decimal point, zero) ]

20:  45, 9  [ RCL 9 ]

21:  45, 8  [ RCL 8 ]

22:  45, 7  [ RCL 7 ]

23:  43, 33, 00 [ GTO 00 ]

// population deviation

24:  45, 1 [ RCL 1 ]

25:  36  [ ENTER ]

26:  36  [ ENTER ]

27:  1  [ 1 ]

28:  30  [ - ]

29:  10  [ ÷ ]

30:  43, 21  [ √ ]

31:  44, 0  [ STO 0 ]

32:  43, 48  [ s ]

33:  45, 0  [ RCL 0 ]

34:  10  [ ÷ ]

35:  34  [ x<>y ]

36:  45, 0  [ RCL 0 ]

37:  10  [ ÷ ]

38:  34  [ x<>y ]

39:  43, 33, 00 [ GTO 00 ]

Example

Fix 6 mode is set.  

Data:

(5.35, 10)

(5,70, 11)

(6.18, 12)

(6.55, 13)

(6.97, 14)

(7.36, 15)

After entering the data points:

[ f ] (PRGM) [ R/S ]

-3.162788 (intercept), [ R↓ ]

2.465287 (slope), [ R↓ ]

0.999396 (correlation, r), [ R↓ ]

1.418000 (covariance)  ( [ R↓ ] to return the stack to it's original configuration )

[ g ] (GTO) 24 [ R/S ]

0.692331 ( σx ) [ x<>y ]

1.707825 ( σy )

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

TI-84 Plus CE: Tracing Polar Equations

TI-84 Plus CE:  Tracing Polar Equations

Traveling Around the Graphs

The programing BLINKING animates a dot around one of four polar equations:

1.  Ellipse

r = b / √(1 - ε^2 * cos^2 θ), where ε = √(a^2 - b^2)/a, a > b

2.  Cardioid

r = a + (1 + cos θ)

3.  3-Leaf Rose

r = a * cos(3 * θ)

4.  Log Spiral

r = a * e^(b * θ)

Notes:

The point alternates through blue (TI-84 Plus CE color 10), green (14), orange (15), and black (12).   The graph itself is light gray (21).  

r1 is the polar equation r1(θ).

Plotting points (not the polar graph) must be in cartesian coordinates (x,y).   

TI-84 Plus CE Program:  BLINKING

"2023-01-15 EWS"

Radian: Polar: FnOff

ZStandard

{10,14,15,12}→L6

Menu("PLOT","ELLIPSE",10,"CARDIOID",11,"3-LEAF ROSE",12,"LOG SPIRAL",13)

Lbl 10

Disp "A≥B"

Prompt A, B

√(A^2-B^2)/A→E

"B/√(1-E^2*cos(θ)^2)"→r1

Goto 20

Lbl 11

Prompt A

"A*(1+cos(θ))"→r1

Goto 20

Lbl 12

Prompt A

"A*cos(3*θ)"→r1

Goto 20

Lbl 13

Disp "A>0, B>0"

Prompt A,B

"A*e^(B*θ)"→r1

Goto 20

Lbl 20

Input "MAX? ",M

Input "NO. STEPS? ",S

M/S→N

FnOn 1

GraphColor(1,21)

0→P: r1(P)→Q

P>Rx(Q,P)→F: P>Ry(Q,P)→G

Pt-On(F,G,1,12)

For(K,1,S)

Wait 0.25

Pt-On(F,G,1,21)

remainder(K,4)+1→C

K*N→P: r1(P)→Q

P>Rx(Q,P)→F: P>Ry(Q,P)→G

Pt-On(F,G,1,L6(C))

End 

Disp "END"

Examples

In all examples, a = 3, b = 2, maximum = 4 * π, steps = 24

Ellipse

Cardioid

 

3-Leaf Rose

Log Spiral

Source

Harris, John W. and Horst Stocker.  Handbook of Mathematics and Computational Science  Springer:  New York.  2006.  ISBN 0-387-94746-9

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Python - Pride and Transgender Flag

Python - Pride and Transgender Flag

The fact that we don't have equality and there are elected officials whose agenda is to destroy civil rights on this planet is ridiculous.  Enough is enough. 

I stand with the LGTBQ community.  Now and forever.   - EWS

Python Code Using Casio MicroPython

Download here:  https://drive.google.com/file/d/1ZSGR_ayEOIwbjsEiNtINCmIhbs7yo0zt/view?usp=share_link

Pride Flag:  pride50.py

# pride flag

from casioplot import *

clear_screen()

for i in range(5,376):

  for k in range(5,35):

    set_pixel(i,k,(228,3,3))

for i in range(5,376):

  for k in range(35,65):

    set_pixel(i,k,(255,140,0))

for i in range(5,376):

  for k in range(65,95):

    set_pixel(i,k,(255,237,0))

for i in range(5,376):

  for k in range(95,125):

    set_pixel(i,k,(0,128,38))

for i in range(5,376):

  for k in range(125,155):

    set_pixel(i,k,(36,64,142))

for i in range(5,376):

  for k in range(155,185):

    set_pixel(i,k,(115,41,130))

show_screen()    

Transgender Flag:  trans50.py

# transgender flag

from casioplot import *

clear_screen()

for i in range(5,376):

  for k in range(5,41):

    set_pixel(i,k,(91,206,250))

for i in range(5,376):

  for k in range(41,77):

    set_pixel(i,k,(245,169,184))

for i in range(5,376):

  for k in range(77,113):

    set_pixel(i,k,(255,255,255))

for i in range(5,376):

  for k in range(113,149):

    set_pixel(i,k,(245,169,184))

for i in range(5,376):

  for k in range(149,185):

    set_pixel(i,k,(91,206,250))

show_screen()    

Python Code Using the Turtle Module from Numworks

Download here:  https://drive.google.com/file/d/1odztGLGfyRsVwq5xSBrfIrc27pj7v58i/view?usp=share_link

Pride Flag:  pride50.nw

from math import *

from turtle import *

hideturtle()

speed(10)

# have to keep size to 1

# want box quality

color(228,3,3)

for i in range(90,60,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(255,140,0)

for i in range(60,30,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(255,237,0)

for i in range(30,0,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

  

color(0,128,38)

for i in range(0,-30,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(36,64,142)

for i in range(-30,-60,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(115,41,130)

for i in range(-60,-90,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

Transgender Flag:  transnw.py

from math import *

from turtle import *

hideturtle()

speed(10)

# have to keep size to 1

# want box quality

color(91,206,250)

for i in range(90,54,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(245,169,184)

for i in range(54,18,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(255,255,255)

for i in range(18,-18,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(245,169,184)

for i in range(-18,-54,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(91,206,250)

for i in range(-54,-90,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

color(91,206,250)

for i in range(90,54,-1):

  penup()

  goto(-140,i)

  pendown()

  forward(280)

Source:

"Pride (Rainbow) Flag Color Codes"  flagcolorcodes Retrieved March 15, 2023.

https://www.flagcolorcodes.com/pride-rainbow

"Transgender Flag Color Codes"   flagcolorcodes.  Retrieved March 15, 2023. 

https://www.flagcolorcodes.com/transgender

Eddie 

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Casio Classpad (fx-CP400): Collection of Functions

Casio Classpad  (fx-CP400):  Collection of Functions

For my birthday post, on Pi (π) Day (March 14), here is a collection of functions for the Casio Classpad (300, 330, fx-CP400, fx-CP500). 

Birthday Probability Function:   bday(c,n)

"The probability that a number in a room do not share a same birthday"

c:  number of categories (i.e. days in a year)

n:  population size

Example:

bday(365, 40):  0.1087681902

Percent Change:  pchg(old, new)

old:  old amount

new:  new amount

Example:

pchg(400,500):  25

Combination with Repetition:  nHr(n,r)

n:  population

r:  number of objects to pick

Example:

nHr(52,5):  3819816

Error Function and Error Compliment Function

Error Function:   erf(x)

Error Compliment Function:  erfc(x)

Note that erf(x) + erfc(x) = 1

Example:

erf(0.60):  0.6038560908

erfc(0.60):  0.3961439092

Law of Cosines

a^2 = b^2 + c^2 - 2*b*c*cos(α)

Finding the Angle α:  cosang(a,b,c)

Finding the Side a:  cosside(b,c,α)

Example:

(Degrees Mode)

a = 20.1, b = 18.5, c = 22.3:  cosang(20.1,18.5,22.3):  58.13961834°  (approx)

b = 24.2, c = 18.9, α = 58°:  cosside(24.2,18.9,58):  21.4032954 (approx)

Fresnel Integrals

Fresnel Cosine Integral:  frescos(u)

C(u) = ∫( cos(π * x^2 ÷ 2) dx, 0, u)

Fresnel Sine Integral:  fressin(u)

S(u) = ∫( sin(π * x^2 ÷ 2) dx, 0, u)

Example:

frescos(1.5):  0.445261176

fressin(1.5):  0.6975049601

Bessel Integral of the First Kind:  bessel(n,x)

J_n(x) = 1 / π * ∫( cos(n * t - x * sin t) dt, 0, π)

Example:

bessel(0,1.5):  0.5118276716

bessel(2,1.2):  0.1593490183

Elliptic Integral of the First Kind:  ellip(x)

K(x) = ∫( 1/ √(1 - x^2 * sin^2 t) dt, 0, π/2)

Example:

ellip(-0.6):  1.750753803

ellip(0.4):  1.639999866

Sine Integral:  Si(x)

Si(x) = ∫( sin t / t dt, 0, x)

Example:

Si(1.8):  1.50581678

Si(6):  1.424687551

Beta Function:  beta(a,b)

β(a, b) = (Γ(a) * Γ(b)) ÷ Γ(a+b)

Example:

beta(2,3): 1/12

beta(1.9,4.6):  0.04470413922 (approx)

Relativity Factor:  relat(v)

factor = √(1 - v^2/c^2)

c = 299792458 m/s

v = velocity

Example:

relat(201E6):  0.7419422153  (approx)

Schwarzschild Radius:  schwarz(m)

r = (2 * G * m)/c^2

G = 6.674E-11  m^3/(kg s^2)

c = 299792458 m/s

m = mass the black hole, kg

r = Schwarzschild Radius, m  (event horizon)

Example:

schwarz(7.89E30):  11717.95418

Distance of a Drop:  dropdist(v0,t)

v0:  initial velocity, m/s

t: time, s

Calculated:  distance, m

Example:

dropdist(15,5):  197.583125

Cycle of a Simple Pendulum:  pendu(l)

l:  length of a string, m

Calulated:  time of the pendulum swing, s

Example:  

pendu(5.5):  4.705446883

Impedance in LRC Series Circuit:  lrcser(R,f,L,C)

R:  resistance, Ω

f:  frequency, Hz

L:  inductance, H

C:  capacity, F

Example:

lrcser(4,80,0.1,50E-6):  11.21437564

Impedance in LRC Parallel Circuit:  lrcpar(R,f,L,C)

R:  resistance, Ω

f:  frequency, Hz

L:  inductance, H

C:  capacity, F

Example:

lrcpar(4,80,0.1,50E-6):  3.999122191

Source for:

Distance of a Drop

Cycle of a Simple Pendulum

Impedance in LRC Series Circuit

Impedance in LRC Parallel Circuit

Scientific Calculator 128 fx-1000F/fx-5000F Owner's Manual.   Casio.  Tokyo, Japan. 

Download the file here:  https://drive.google.com/file/d/1M54HlJ9dP95VBEmGzkUozGzJxKGpiHMh/view?usp=share_link

Eddie 

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